The fourier-analysis tag has no wiki summary.

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### Problem with operator and Fourier transform

I am currently dealing with a problem in functional analysis where I want to show the following.
Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$
if ...

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208 views

### Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...

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29 views

### Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces.
In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$
Take $f(x)= ...

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97 views

### Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that
$T(x):= \sum_{n \in \mathbb{Z}} ...

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46 views

### $A_n \not \rightharpoonup A$ in $L_1[-\pi; \pi] $ ( $A_n$ is partial fourier sum )

Let
\begin{equation*}
(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) + b_k sin(kt), \\
a_k = \frac{1}{\sqrt{\pi}} \int_{-\pi}^{\pi} x(t) cos(kt) dt, \\
b_k = \frac{1}{\sqrt{\pi}} ...

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117 views

### Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent (“weakly”)?

It is wellknown that there is a convergence in norm for Fourier series in $L_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I ...

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87 views

### Transformation of kernel

I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if ...

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177 views

### Variations on the Mellin and Dirichlet transforms

There are a number of variations on the Laplace transform that turn up all over math. Some examples:
$\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
$\sum_{-\infty}^{\infty} ...

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336 views

### Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...

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**1**answer

247 views

### Bandwidth approximation for a nonlinear problem

Can anyone please help me with this problem.
I must let you know from the beginning that it's not an easy one.
"Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...

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464 views

### Is the integral always nonzero?

Let
$$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$
where
$$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < ...

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61 views

### prove that a function is approximatively three dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$.
Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...

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136 views

### Largest Fourier coefficient of sparse boolean function

Consider a Boolean function $f: GF(2)^n \rightarrow \{0, 1 \}$. I would like to show that if $f$ is sparse, i.e. $\sum f(i) \leq t$, then $f$ must have a large Fourier coefficient. (A Fourier ...

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**1**answer

580 views

### Constructing an L2-space with a given orthonormal basis

Coming from a completely different world, I am trying to learn some very basic Fourier analysis, and have been scratching my head around this (it may be a very stupid question with an obvious answer):
...

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323 views

### Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $j'_{n+1/2,1}$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$.
...

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### Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...

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### Nonlinear Schrödinger blow-up for non radial solutions

I am studying a paper of Frank Merle and Pierre Raphaël,
http://math.unice.fr/~praphael/Publications/blow-up-norme-critique.pdf.
The equations are
$$
i\partial_tu+\Delta u=-|u|^{p-1}u
$$
on ...

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**1**answer

205 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

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26 views

### Gaussian gabor frame

It is widely known that $\phi(x)=e^{-\frac{x^2}{2}}$ does not define a Gabor frame if we consider translations by units of $1$ and multiplication by $e^{2 \pi inx}$for $n \in \mathbb{N}.$ A way to ...

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### Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$.
Define a sequence of banded ...

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113 views

### Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...

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251 views

### Is there a name for this space?

I'm just asking if there is a name for the space of functions on $\mathbb R^n$ whose norm is defined by
$$ \|f\|=\|\hat f\|_{L^p} $$
for $p\in [1,\infty]$. I find it handy to give it a name when ...

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153 views

### Fourier coefficients of real analytic functions on an n-dimension torus

Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of ...

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### Reference request : Besov spaces on ubounded domains

As I am relatively new to these matters, I would like to know if you could provide me a reference for Besov spaces on unbounded domains, because when I checked the first tome of Triebel's Theory of ...

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115 views

### Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS

Using Hardy-Littlewood-Sobolev inequality, we can prove that:
$$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C ...

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### Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

[I have asked this question on S.E. M; but I have not got any answer; and hope this is o.k. for M.O]
Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and ...

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### Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...

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### How is the structure of spectrum in cap-sets with no strong increments unrealistic if density is too large?

I am reading this excellent paper by Bateman and Katz on improved bounds on the cap set problem.
Let A be a set in $\mathbb{F}_3^n$ containing no 3-term arithmetic progression and let $A(x)$ denote ...

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87 views

### On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$):
$$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...

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128 views

### Determining the Fourier transform

Let $d>2$. Let $M$ be a 2-dimensional submanifold of $\mathbb{R}^d$. For instance (and this is the type of example I primarily care about) we could have $M$ being the set of scalar multiples of a ...

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57 views

### separating two parameters in an oscillatory integral

Consider the following oscillatory integral with two parameters: $$I(a,b)=\int_{\mathbb{R}}e^{i(ax^2+bx)}\psi(x)\,dx$$ where $\psi$ is smooth and supported in $\{x:|x|\in[1/2,2]\}$.
Can we write ...

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**1**answer

77 views

### Estimate self crossings of a curve parameterized by a trigonometric polynomial

Given z on the unit circle, let $P(z)= \sum\limits_{k=-D}^D p_k z^k $.
Can one estimate the number of self crossings of the following curve with an analytic expression in terms of the coefficients ...

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276 views

### Frequency of a representation of SO(3)

When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In ...

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380 views

### Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space ...

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143 views

### Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than ...

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86 views

### Summing a function at integer points

For a real function $f$ defined on the reals, and real $y$, define the 2-way infinite sum
$$ F_f(y) = \sum_{i\in\mathbb Z} f(y+i). $$
If $F_f(y)$ is defined for all $y$, it is periodic of period 1.
...

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### Condition for boundedness in stationary phase theorem

I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...

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### How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?

We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$
By Helson-Kahane-Katznelson-Rudin Theorem, it follows that,
"Let $F$ be a function on $\mathbb C$ and if ...

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418 views

### Basic examples of induction on scales arguments

An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth.
Here is a ...

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212 views

### On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$
with $s_0$ a zero of the Riemann Zeta function in the critical strip.
This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...

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### Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
...

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### Distribution of Fourier coefficients of Maass forms

In the sense of Maass an automorphic function $\phi$ with Laplace-Beltrami eigenvalue $\frac{(d-1)^2}{4}+t^2$ on $d$-dimensional hyperbolic space which can be thought as ...

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138 views

### Is there a Poisson Summation formula for imprimitive Dirichlet characters?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ?
For a primitive Dirichlet character $\chi$ we have:
...

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111 views

### Uniform $L^p-L^{p'}$ bound of a Fourier multiplier

Let $(\tau,\xi)\in\mathbb{R}\times \mathbb{R}^n$, and consider the function
$$
m_{\epsilon}(\tau,\xi)=\frac{1}{\tau+|\xi|^4+\epsilon|\xi|^2+i}
$$
in $\mathbb{R}^{n+1}$. My first question is that does ...

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296 views

### What is the Fourier transform of this function?

Consider the function
$$
f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du.
$$
It is known that $f(x_1,x_2)\in ...

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### Number of critical points of smooth functions on $S^1$

Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?

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321 views

### Conditions for positivity of Fourier transform

Assume you are given a non-negative, continuous, radial function $f\in L^q(\mathbb{R}^3)$ (for any $q\geq 1$).
Are there any conditions which would guarantee that the Fourier transform of $f$, that ...

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76 views

### variation norm of a Fourier transform

Motivated by certain uniform estimate in oscillatory integrals, I am now trying to calculate the Fourier transform of the function ${\large e^{i|t|^{\epsilon}}/t}$ on $\mathbb{R}$, where $\epsilon\in ...

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### Abstract stationary phase

I have been reading Semi-classical analysis by Guillemin and Sternberg. At the end of Chapter 8, they gave an abstract version of the stationary phase method. I have a hard time figuring out what ...

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204 views

### A calculus question related to the nonnegative definite functions

I am looking for some sufficient conditions for an even, continuous, nonnegative, non increasing function $f(x)$ on $R$ such that
$$
\int_0^\infty \cos(xz) f(z) d z \ge 0 \qquad\text{for all $x\ge ...